In all the theorems above, the existence of the metric \(h\)  comes from ampleness of \(E\) , and the existence of metrics \(g\)  can be deduced easily as explained. So the real assumptions lie in \(\Omega \) , \(M\) , and the inequalities they have to satisfy. To weaken or remove these inequalities, one possible direction is to use geometric flows as in [1]}, [2]}, [3]}, [4]}. Another possible direction is to use the interplay between the optimal \(L^2\) -estimates and the positivity of curvature (see [5]}, [6]}, [7]}, [8]}, [9]}).
